Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reflections on Conformal Spectra

Hyungrok Kim, Petr Kravchuk, Hirosi Ooguri

TL;DR

The paper uses modular invariance and crossing symmetry to uncover approximate reflection relations in the spectra of 2D partition functions and higher-dimensional four-point functions in key scaling limits (large c, large Δ0, and large d). By reformulating spectral data as polynomial moments and applying linear programming duality, the authors derive tail bounds and Cardy-like growth for the averaged density and OPE coefficients, with universal features tied to a sparse light spectrum. They develop both exact and finite-Δ0 bounds, and show how saddle-point dominance explains the observed universality while also highlighting constraints that persist beyond the heavy-spectrum regime. The work offers a framework that could inform numerical bootstrap strategies and clarifies how crossing and modular constraints shape the heavy part of the spectrum across dimensions and limits.

Abstract

We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-point function in any dimension in the limit of large scaling dimensions $Δ_0$ of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite $Δ_0$ as well as for large $Δ_0$. We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.

Reflections on Conformal Spectra

TL;DR

The paper uses modular invariance and crossing symmetry to uncover approximate reflection relations in the spectra of 2D partition functions and higher-dimensional four-point functions in key scaling limits (large c, large Δ0, and large d). By reformulating spectral data as polynomial moments and applying linear programming duality, the authors derive tail bounds and Cardy-like growth for the averaged density and OPE coefficients, with universal features tied to a sparse light spectrum. They develop both exact and finite-Δ0 bounds, and show how saddle-point dominance explains the observed universality while also highlighting constraints that persist beyond the heavy-spectrum regime. The work offers a framework that could inform numerical bootstrap strategies and clarifies how crossing and modular constraints shape the heavy part of the spectrum across dimensions and limits.

Abstract

We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-point function in any dimension in the limit of large scaling dimensions of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite as well as for large . We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.

Paper Structure

This paper contains 17 sections, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Partition function
  4. Four-point function
  5. Finite-$\Delta_0$ bounds
  6. Cardy formula
  7. Four-point function in large spacetime dimension
  8. Derivations
  9. Modular Invariance
  10. Reflection Symmetry
  11. Bound on Tail
  12. Crossing Symmetry
  13. Reflection and Bounds
  14. Cardy formula
  15. Discussion
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Allowed range for $\Lambda_x$ as a function of $x$ for $\delta_0=1$ and $\delta_0=1/2$.
  • Figure 2: Plots of $f(x)/\tilde{f}(x)$ for different orders of approximation and values of angular momenta in four dimensions.
  • Figure 3: Plot of $f(x)/\tilde{f}(x)$ for different orders of approximation and the value of angular momenta at the unitarity bound in four dimensions.